I'd like to add that the TDK (Türk Dİl Kurumu) dictionary defines the phrase "çıbanın başını koparmak" as "ağır bir sorunun patlak vermesine yol açmak" (http://tdkterim.gov.tr/bts/)
So it refers to actions rather than to people and therefore could be more or less the exact equivalent of the phrase "the thin end of the wedge".